3. Let D: R[x] ® R[x] be the
derivative map
given by D(a0+a1x+a2x2+ ¼anxn) = a1+2a2x+ ¼+nanxn-1.
Show that D is not a homomorphism. (Hint: Compute D(x2).)

4. Suppose that R and S are rings, R' is a subring of R,
and S' is a subring of S. Show that R' ×S' is a subring of
R ×S.

5. Show that Z6× Z5@ Z10× Z3.

6. Let R be a ring with identity. An element e Î R
is called idempotent if e2 = e. The elements 0 and 1 are
called the trivial idempotents of R. All other idempotents
(if any exist) are called nontrivial idempotents